Trialities of orthosymplectic $\mathcal{W}$-algebras

Abstract

Trialities of $\mathcal{W}$-algebras are isomorphisms between the affine cosets of three different $\mathcal{W}$-(super)algebras, and were first conjectured in the physics literature by Gaiotto and Rapčák. In this paper we prove trialities among eight families of $\mathcal{W}$-(super)algebras of types $B$, $C$, and $D$. The key idea is to identify the affine cosets of these algebras with one-parameter quotients of the universal two-parameter even spin ${\mathcal{W}}_{\infty }$-algebra which was recently constructed by Kanade and the second author. Our result is a vast generalization of both Feigin-Frenkel duality in types $B$, $C$, and $D$, and the coset realization of principal $\mathcal{W}$-algebras of type $D$ due to Arakawa and us. It also provides a new coset realization of principal $\mathcal{W}$-algebras of types $B$ and $C$. As an application, we prove the rationality of the affine vertex superalgebra $L_k({\mathfrak{osp}}_{1|2n})$, the minimal $\mathcal{W}$-algebra ${\mathcal{W}}_{k-1/2}({\mathfrak{sp}}_{2n+2},f_{\text{min}})$, and the coset $\text{Com}(L_k({\mathfrak{sp}}_{2m}),L_k({\mathfrak{sp}}_{2n}))$, for all integers $k,n,m\ge 1$ with $m<n$. We also prove the rationality of some families of principal $\mathcal{W}$-superalgebras of ${\mathfrak{osp}}_{1|2n}$ and ${\mathfrak{osp}}_{2|2n}$, and subregular $\mathcal{W}$-algebras of ${\mathfrak{so}}_{2n+1}$

arXiv commentSome corrections and expository improvements, references added, final version to appear in Advances in Mathematics

References in corpus (40)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Quantum Reduction for Affine Superalgebras
• Triality in Minimal Model Holography
• W-symmetry, topological vertex and affine Yangian
• Braided tensor categories and extensions of vertex operator algebras
• Integrable highest weight modules over affine superalgebras and Appell's function
• W-algebras as coset vertex algebras
• Schur-Weyl Duality for Heisenberg Cosets
• Rationality of W-algebras: principal nilpotent cases
• Rationality of admissible affine vertex algebras in the category O
• $W_n^{(2)}$ algebras
• Cosets of affine vertex algebras inside larger structures
• Cohomological Hall algebras, vertex algebras and instantons
• Simple current extensions beyond semi-simplicity
• Braided tensor categories of admissible modules for affine Lie algebras
• Exploring ${\mathcal{W}}_{\infty }$ in the quadratic basis
• Conformal embeddings of affine vertex algebras in minimal $W$-algebras I: structural results
• Trialities of $\mathcal{W}$-algebras
• Duality of subregular W-algebras and principal W-superalgebras
• A logarithmic generalization of tensor product theory for modules for a vertex operator algebra
• On the tensor structure of modules for compact orbifold vertex operator algebras
• Orbifolds and cosets of minimal $\mathcal{W}$-algebras
• Screening operators for W-algebras
• Gluing vertex algebras
• Quantum Supergroups III. Twistors
• Universal two-parameter ${\mathcal{W}}_{\infty }$-algebra and vertex algebras of type $\mathcal{W}(2,3,\dots ,N)$
• Tensor categories of affine Lie algebras beyond admissible levels
• Invariant theory and the Heisenberg vertex algebra
• Orbifolds of symplectic fermion algebras
• Universal two-parameter even spin ${\mathcal{W}}_{\infty }$-algebra
• S-duality for the large $N=4$ superconformal algebra
• Superconformal minimal models and admissible Jack polynomials
• Correspondences of categories for subregular W-algebras and principal W-superalgebras
• $N=4$ superconformal algebras and diagonal cosets
• On exceptional nilpotents in semisimple Lie algebras
• Rationality of the exceptional W-algebras ${\mathcal{W}}_k({\mathfrak{sp}}_4,f_{subreg})$ associated with subregular nilpotent elements of ${\mathfrak{sp}}_4$
• Generalized parafermions of orthogonal type
• Higher rank FZZ-dualities
• Symplectic level-rank duality via tensor categories
• Quantum dimensions and irreducible modules of some diagonal coset vertex operator algebras

Cited by in corpus (12)

• Quantum toroidal algebras and solvable structures in gauge/string theory
• Boundary vertex algebras for 3d $\mathcal{N}=4$ rank-0 SCFTs
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• Singularities of nilpotent Slodowy slices and collapsing levels of W-algebras
• Cosets of free field algebras via arc spaces
• Ordinary modules for vertex algebras of ${\mathfrak{osp}}_{1|2n}$
• $N=1$ super Virasoro tensor categories
• Cosets from equivariant W-algebras
• Orthosymplectic Feigin-Semikhatov duality
• Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras
• Feigin-Semikhatov conjecture and related topics
• Duality via convolution of W-algebras